smile.clustering

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Clustering Large Applications based upon RANdomized Search. CLARANS is an efficient medoid-based clustering algorithm. The k-medoids algorithm is an adaptation of the k-means algorithm. Rather than calculate the mean of the items in each cluster, a representative item, or medoid, is chosen for each cluster at each iteration. In CLARANS, the process of finding k medoids from n objects is viewed abstractly as searching through a certain graph. In the graph, a node is represented by a set of k objects as selected medoids. Two nodes are neighbors if their sets differ by only one object. In each iteration, CLARANS considers a set of randomly chosen neighbor nodes as candidate of new medoids. We will move to the neighbor node if the neighbor is a better choice for medoids. Otherwise, a local optima is discovered. The entire process is repeated multiple time to find better.

CLARANS has two parameters: the maximum number of neighbors examined (maxNeighbor) and the number of local minima obtained (numLocal). The higher the value of maxNeighbor, the closer is CLARANS to PAM, and the longer is each search of a local minima. But the quality of such a local minima is higher and fewer local minima needs to be obtained.

====References:====

• R. Ng and J. Han. CLARANS: A Method for Clustering Objects for Spatial Data Mining. IEEE TRANS. KNOWLEDGE AND DATA ENGINEERING, 2002.

Deterministic annealing clustering. Deterministic annealing extends soft-clustering to an annealing process. For each temperature value, the algorithm iterates between the calculation of all posteriori probabilities and the update of the centroids vectors, until convergence is reached. The annealing starts with a high temperature. Here, all centroids vectors converge to the center of the pattern distribution (independent of their initial positions). Below a critical temperature the vectors start to split. Further decreasing the temperature leads to more splittings until all centroids vectors are separate. The annealing can therefore avoid (if it is sufficiently slow) the convergence to local minima.

====References:====

• Kenneth Rose. Deterministic Annealing for Clustering, Compression, Classification, Regression, and Speech Recognition.

Density-Based Spatial Clustering of Applications with Noise. DBSCAN finds a number of clusters starting from the estimated density distribution of corresponding nodes.

DBSCAN requires two parameters: radius (i.e. neighborhood radius) and the number of minimum points required to form a cluster (minPts). It starts with an arbitrary starting point that has not been visited. This point's neighborhood is retrieved, and if it contains sufficient number of points, a cluster is started. Otherwise, the point is labeled as noise. Note that this point might later be found in a sufficiently sized radius-environment of a different point and hence be made part of a cluster.

If a point is found to be part of a cluster, its neighborhood is also part of that cluster. Hence, all points that are found within the neighborhood are added, as is their own neighborhood. This process continues until the cluster is completely found. Then, a new unvisited point is retrieved and processed, leading to the discovery of a further cluster of noise.

DBSCAN visits each point of the database, possibly multiple times (e.g., as candidates to different clusters). For practical considerations, however, the time complexity is mostly governed by the number of nearest neighbor queries. DBSCAN executes exactly one such query for each point, and if an indexing structure is used that executes such a neighborhood query in O(log n), an overall runtime complexity of O(n log n) is obtained.

DBSCAN has many advantages such as

• DBSCAN does not need to know the number of clusters in the data a priori, as opposed to k-means.
• DBSCAN can find arbitrarily shaped clusters. It can even find clusters completely surrounded by (but not connected to) a different cluster. Due to the MinPts parameter, the so-called single-link effect (different clusters being connected by a thin line of points) is reduced.
• DBSCAN has a notion of noise.
• DBSCAN requires just two parameters and is mostly insensitive to the ordering of the points in the database. (Only points sitting on the edge of two different clusters might swap cluster membership if the ordering of the points is changed, and the cluster assignment is unique only up to isomorphism.)

On the other hand, DBSCAN has the disadvantages of

• In high dimensional space, the data are sparse everywhere because of the curse of dimensionality. Therefore, DBSCAN doesn't work well on high-dimensional data in general.
• DBSCAN does not respond well to data sets with varying densities.

====References:====

• Martin Ester, Hans-Peter Kriegel, Jorg Sander, Xiaowei Xu (1996-). A density-based algorithm for discovering clusters in large spatial databases with noise". KDD, 1996.
• Jorg Sander, Martin Ester, Hans-Peter Kriegel, Xiaowei Xu. (1998). Density-Based Clustering in Spatial Databases: The Algorithm GDBSCAN and Its Applications. 1998.

Density-Based Spatial Clustering of Applications with Noise. DBSCAN finds a number of clusters starting from the estimated density distribution of corresponding nodes.

DBSCAN with Euclidean distance. DBSCAN finds a number of clusters starting from the estimated density distribution of corresponding nodes.

DENsity CLUstering. The DENCLUE algorithm employs a cluster model based on kernel density estimation. A cluster is defined by a local maximum of the estimated density function. Data points going to the same local maximum are put into the same cluster.

Clearly, DENCLUE doesn't work on data with uniform distribution. In high dimensional space, the data always look like uniformly distributed because of the curse of dimensionality. Therefore, DENCLUDE doesn't work well on high-dimensional data in general.

====References:====

• A. Hinneburg and D. A. Keim. A general approach to clustering in large databases with noise. Knowledge and Information Systems, 5(4):387-415, 2003.
• Alexander Hinneburg and Hans-Henning Gabriel. DENCLUE 2.0: Fast Clustering based on Kernel Density Estimation. IDA, 2007.

G-Means clustering algorithm, an extended K-Means which tries to automatically determine the number of clusters by normality test. The G-means algorithm is based on a statistical test for the hypothesis that a subset of data follows a Gaussian distribution. G-means runs k-means with increasing k in a hierarchical fashion until the test accepts the hypothesis that the data assigned to each k-means center are Gaussian.

====References:====

• G. Hamerly and C. Elkan. Learning the k in k-means. NIPS, 2003.

Agglomerative Hierarchical Clustering. This method seeks to build a hierarchy of clusters in a bottom up approach: each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy. The results of hierarchical clustering are usually presented in a dendrogram.

In general, the merges are determined in a greedy manner. In order to decide which clusters should be combined, a measure of dissimilarity between sets of observations is required. In most methods of hierarchical clustering, this is achieved by use of an appropriate metric, and a linkage criteria which specifies the dissimilarity of sets as a function of the pairwise distances of observations in the sets.

Hierarchical clustering has the distinct advantage that any valid measure of distance can be used. In fact, the observations themselves are not required: all that is used is a matrix of distances.

== References ==

• David Eppstein. Fast hierarchical clustering and other applications of dynamic closest pairs. SODA 1998.

Agglomerative Hierarchical Clustering. This method seeks to build a hierarchy of clusters in a bottom up approach: each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy. The results of hierarchical clustering are usually presented in a dendrogram.

In general, the merges are determined in a greedy manner. In order to decide which clusters should be combined, a measure of dissimilarity between sets of observations is required. In most methods of hierarchical clustering, this is achieved by use of an appropriate metric, and a linkage criteria which specifies the dissimilarity of sets as a function of the pairwise distances of observations in the sets.

Hierarchical clustering has the distinct advantage that any valid measure of distance can be used. In fact, the observations themselves are not required: all that is used is a matrix of distances.

== References ==

• David Eppstein. Fast hierarchical clustering and other applications of dynamic closest pairs. SODA 1998.

K-Means clustering. The algorithm partitions n observations into k clusters in which each observation belongs to the cluster with the nearest mean. Although finding an exact solution to the k-means problem for arbitrary input is NP-hard, the standard approach to finding an approximate solution (often called Lloyd's algorithm or the k-means algorithm) is used widely and frequently finds reasonable solutions quickly.

However, the k-means algorithm has at least two major theoretic shortcomings:

• First, it has been shown that the worst case running time of the algorithm is super-polynomial in the input size.
• Second, the approximation found can be arbitrarily bad with respect to the objective function compared to the optimal learn.

In this implementation, we use k-means++ which addresses the second of these obstacles by specifying a procedure to initialize the cluster centers before proceeding with the standard k-means optimization iterations. With the k-means++ initialization, the algorithm is guaranteed to find a solution that is O(log k) competitive to the optimal k-means solution.

We also use k-d trees to speed up each k-means step as described in the filter algorithm by Kanungo, et al.

K-means is a hard clustering method, i.e. each sample is assigned to a specific cluster. In contrast, soft clustering, e.g. the Expectation-Maximization algorithm for Gaussian mixtures, assign samples to different clusters with different probabilities.

====References:====

• Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Christine D. Piatko, Ruth Silverman, and Angela Y. Wu. An Efficient k-Means Clustering Algorithm: Analysis and Implementation. IEEE TRANS. PAMI, 2002.
• D. Arthur and S. Vassilvitskii. "K-means++: the advantages of careful seeding". ACM-SIAM symposium on Discrete algorithms, 1027-1035, 2007.
• Anna D. Peterson, Arka P. Ghosh and Ranjan Maitra. A systematic evaluation of different methods for initializing the K-means clustering algorithm. 2010.

This method runs the algorithm for given times and return the best one with smallest distortion.

K-Modes clustering. K-Modes is the binary equivalent for K-Means. The mean update for centroids is replace by the mode one which is a majority vote among element of each cluster.

Nonparametric Minimum Conditional Entropy Clustering. This method performs very well especially when the exact number of clusters is unknown. The method can also correctly reveal the structure of data and effectively identify outliers simultaneously.

The clustering criterion is based on the conditional entropy H(C | x), where C is the cluster label and x is an observation. According to Fano's inequality, we can estimate C with a low probability of error only if the conditional entropy H(C | X) is small. MEC also generalizes the criterion by replacing Shannon's entropy with Havrda-Charvat's structural α-entropy. Interestingly, the minimum entropy criterion based on structural α-entropy is equal to the probability error of the nearest neighbor method when α= 2. To estimate p(C | x), MEC employs Parzen density estimation, a nonparametric approach.

MEC is an iterative algorithm starting with an initial partition given by any other clustering methods, e.g. k-means, CLARNAS, hierarchical clustering, etc. Note that a random initialization is NOT appropriate.

====References:====

• Haifeng Li. All rights reserved., Keshu Zhang, and Tao Jiang. Minimum Entropy Clustering and Applications to Gene Expression Analysis. CSB, 2004.

Nonparametric Minimum Conditional Entropy Clustering.

Nonparametric Minimum Conditional Entropy Clustering. Assume Euclidean distance.

Nonparametric Minimum Conditional Entropy Clustering.

The Sequential Information Bottleneck algorithm. SIB clusters co-occurrence data such as text documents vs words. SIB is guaranteed to converge to a local maximum of the information. Moreover, the time and space complexity are significantly improved in contrast to the agglomerative IB algorithm.

In analogy to K-Means, SIB's update formulas are essentially same as the EM algorithm for estimating finite Gaussian mixture model by replacing regular Euclidean distance with Kullback-Leibler divergence, which is clearly a better dissimilarity measure for co-occurrence data. However, the common batch updating rule (assigning all instances to nearest centroids and then updating centroids) of K-Means won't work in SIB, which has to work in a sequential way (reassigning (if better) each instance then immediately update related centroids). It might be because K-L divergence is very sensitive and the centroids may be significantly changed in each iteration in batch updating rule.

Note that this implementation has a little difference from the original paper, in which a weighted Jensen-Shannon divergence is employed as a criterion to assign a randomly-picked sample to a different cluster. However, this doesn't work well in some cases as we experienced probably because the weighted JS divergence gives too much weight to clusters which is much larger than a single sample. In this implementation, we instead use the regular/unweighted Jensen-Shannon divergence.

====References:====

• N. Tishby, F.C. Pereira, and W. Bialek. The information bottleneck method. 1999.
• N. Slonim, N. Friedman, and N. Tishby. Unsupervised document classification using sequential information maximization. ACM SIGIR, 2002.
• Jaakko Peltonen, Janne Sinkkonen, and Samuel Kaski. Sequential information bottleneck for finite data. ICML, 2004.

Spectral Clustering. Given a set of data points, the similarity matrix may be defined as a matrix S where S_ij represents a measure of the similarity between points. Spectral clustering techniques make use of the spectrum of the similarity matrix of the data to perform dimensionality reduction for clustering in fewer dimensions. Then the clustering will be performed in the dimension-reduce space, in which clusters of non-convex shape may become tight. There are some intriguing similarities between spectral clustering methods and kernel PCA, which has been empirically observed to perform clustering.

====References:====

• A.Y. Ng, M.I. Jordan, and Y. Weiss. On Spectral Clustering: Analysis and an algorithm. NIPS, 2001.
• Marina Maila and Jianbo Shi. Learning segmentation by random walks. NIPS, 2000.
• Deepak Verma and Marina Meila. A Comparison of Spectral Clustering Algorithms. 2003.

Spectral clustering.

Spectral clustering with Nystrom approximation.

X-Means clustering algorithm, an extended K-Means which tries to automatically determine the number of clusters based on BIC scores. Starting with only one cluster, the X-Means algorithm goes into action after each run of K-Means, making local decisions about which subset of the current centroids should split themselves in order to better fit the data. The splitting decision is done by computing the Bayesian Information Criterion (BIC).

====References:====

• Dan Pelleg and Andrew Moore. X-means: Extending K-means with Efficient Estimation of the Number of Clusters. ICML, 2000.